Projective Ring Line of an Arbitrary Single Qudit

TL;DR

This paper explores the algebraic and geometric structure of single qudits of any dimension, revealing intricate relations between their Pauli groups and projective lines over modular rings, with explicit formulas and structural insights.

Contribution

It provides a novel algebraic geometric analysis of single qudits for arbitrary dimensions, linking symplectic modules to projective lines over rings and uncovering new structural properties.

Findings

Explicit formulas for commuting Pauli operators and projective line points.

Perp-sets are not unions of projective line points unless dimension is a product of distinct primes.

Operators form disjoint layers based on vector degrees.

Abstract

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it any} positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring $\bZ_{d}$. Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of $\bZ^{2}_{d}$. We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless $d$ is a product of distinct primes. The operators are also seen to be structured into disjoint `layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.